Let a1, a2, a3,……… an, be n positive consecutive terms of an arithmetic progression If d

Engineering

Mathematics

Methods to Evaluate Limits

Question

Let a₁, a₂, a₃,……… a_n, be n positive consecutive terms of an arithmetic progression If d > 0 is its common difference, then

$\lim_{n \to \infty} \sqrt{\frac{d}{n}} (\frac{1}{\sqrt{a_{1}} + \sqrt{a_{2}}} + ...... + \frac{1}{\sqrt{a_{n - 1}} + \sqrt{a_{n}}})$ is

$\sqrt{d}$

$\frac{1}{\sqrt{d}}$

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$(\frac{1}{\sqrt{a_{1}} + \sqrt{a_{2}}} + \frac{1}{\sqrt{a_{2}} + \sqrt{a_{3}}} + . ....... + \frac{1}{\sqrt{a_{n - 1}} + \sqrt{a_{n}}})$

= $(\frac{\sqrt{a_{2}} - \sqrt{a_{1}}}{a_{2} - a_{1}} + \frac{\sqrt{a_{3}} - \sqrt{a_{2}}}{a_{3} - a_{2}} + \frac{\sqrt{a_{4}} - \sqrt{a_{3}}}{a_{4} - a_{3}} ..... + \frac{\sqrt{a_{n}} - \sqrt{a_{n - 1}}}{a_{n} - a_{n - 1}})$

= $\frac{1}{d} (\sqrt{a_{n}} - \sqrt{a_{1}})$

∴ $\underset{n \to \infty}{Lt} \sqrt{\frac{d}{n}} (\frac{1}{\sqrt{a_{1}} + \sqrt{a_{2}}} + \frac{1}{\sqrt{a_{2}} + \sqrt{a_{3}}} + . ..... + \frac{1}{\sqrt{a_{n - 1}} + \sqrt{a_{n}}})$

= $\underset{n \to \infty}{Lt} \sqrt{\frac{d}{n}} (\frac{\sqrt{a_{n}} - \sqrt{a_{1}}}{d})$

= $\underset{n \to \infty}{Lt} (\frac{\sqrt{a_{1} + (n - 1) d} - \sqrt{a_{1}}}{\sqrt{n} \sqrt{d}})$ = $\underset{n \to \infty}{Lt} (\sqrt{\frac{a_{1}}{nd} + (1 - \frac{1}{n})} - \sqrt{\frac{a_{1}}{nd}})$

= 1